On the marginal instability of linear switched systems

Stability properties for continuous-time linear switched systems are determined by the Lyapunov exponent associated with the system, which is the analogous of the joint spectral radius for the discrete-time case. This paper is concerned with the characterizations of stability properties when the Lyapunov exponent is zero. In this case it is well known that the system can be stable as well as unstable, even if it is never asymptotically stable nor it admits a trajectory blowing up exponentially fast. Our main result asserts that a switched system whose Lyapunov exponent is zero may be unstable only if a certain resonance condition is satisfied.

[1]  Yacine Chitour,et al.  Common Polynomial Lyapunov Functions for Linear Switched Systems , 2006, SIAM J. Control. Optim..

[2]  L. Gurvits.  Stability of discrete linear inclusion , 1995 .

[3]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[4]  J. Bell.  A gap result for the norms of semigroups of matrices , 2005 .

[5]  H. Sussmann,et al.  Continuous Dependence with Respect to the Input of Trajectories of Control-Affine Systems , 1999 .

[6]  Vincent D. Blondel,et al.  An Elementary Counterexample to the Finiteness Conjecture , 2002, SIAM J. Matrix Anal. Appl..

[7]  Zhendong Sun,et al.  A Note on Marginal Stability of Switched Systems , 2007, IEEE Transactions on Automatic Control.

[8]  U. Boscain.  Stability of planar switched systems: the linear single, input case , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[9]  J. Tsitsiklis,et al.  The boundedness of all products of a pair of matrices is undecidable , 2000 .

[10]  C. Curtis,et al.  Representation theory of finite groups and associated algebras , 1962 .

[11]  Paolo Mason,et al.  A note on stability conditions for planar switched systems , 2009, Int. J. Control.

[12]  N. Barabanov.  Absolute characteristic exponent of a class of linear nonstatinoary systems of differential equations , 1988 .

[13]  R. Jungers.  The Joint Spectral Radius: Theory and Applications , 2009 .

[14]  Владимир Юрьевич Протасов,et al.  Фрактальные кривые и всплески@@@Fractal curves and wavelets , 2006 .

[15]  Robert Shorten,et al.  Stability Criteria for Switched and Hybrid Systems , 2007, SIAM Rev..

[16]  F. Wirth.  The generalized spectral radius and extremal norms , 2002 .

[17]  Rapha L. Jungers.  The Joint Spectral Radius , 2009 .

[18]  N. Barabanov,et al.  Asymptotic behavior of extremal solutions and structure of extremal norms of linear differential inclusions of order three , 2008 .

[19]  Michael Margaliot,et al.  Stability analysis of switched systems using variational principles: An introduction , 2006, Autom..

[20]  Vincent D. Blondel,et al.  Efficient algorithms for deciding the type of growth of products of integer matrices , 2006, ArXiv.